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OPTIMAL OPERATING POLICIES OF AN EPQ

MODEL WITH STOCK DEPENDENT PRODUCTION

RATE AND TIME DEPENDENT DEMAND HAVING

PARETO DECAY

Dr.Y.Srinivasa Rao1, K.Srinivasa Rao

2, Dr.V.V.S.Kesava Rao

3

1Professor, Avanthi’s St.Theressa Inst. of Engg. and Tech., Garividi, Vizianagaram (India)

2,3Professor, Department of Statistics Andhra University Visakhapatnam (India)

ABSTRACT

In this paper an economic production quantity model is developed and analyzed for deteriorating items. Here it is

assumed that the production rate is dependent on stock on-hand having Pareto rate of decay. It is further assumed

that the demand follows a power pattern with an index parameter. The model behavior is analyzed by deriving the

instantaneous state of inventory at time t, stock loss due to deterioration and production quantity. By minimizing the

total cost of production the optimal values of production downtime (time point at which production stops),

production uptime (time point at which production resumes) and optimal production quantity are derived. The

sensitivity analysis of the model revealed that the stock dependent production rate can reduce the total production

cost and unnecessary inventory of goods. It is also observed that the Pareto rate of decay can well characterize the

deterioration of a commodity like cement. This model also includes several of the earlier models as particular cases.

Keywords EPQ, Pareto decay, stock dependent production, power demand pattern

I. INTRODUCTION

Recently much emphasis is given for analyzing economic production quantity models which provide the basic frame

work for monitoring and controlling the production processes for deteriorating items like cement, oil, food

processing, ic chips, textiles etc. Goyal and Giri [6], Ruxian Li et al. [10], Pentico and Drake [9] have reviewed

several types of inventory models for deteriorating items. Deterioration is characterized as loss of life or obsolete or

decay or evaporate due to various facto` Since the deterioration is influenced by several random factors, the life time

of the commodity is considered as random (Nahmias [8]). Several authors develop various economic production

quantity models for deteriorating items with various assumptions on life time of the commodity, pattern of demand

and production. (Skouri and Papachristos [11], Chen and Chen [4], Balkhi [2], Manna and Chaudhari [7], Teng et al.

[17], Abad [1], Goyal and Giri [5]) have developed inventory models for deteriorating items with finite rate of

production. Recently Srinivasa Rao et al. [13], Srinivasa Rao et al. [15], Begum et al. [3] have developed and

analyzed EPQ models for deteriorating items with generalized Pareto decay and constant rate of production.

Umamaheswara Rao et al. [18], Venkata Subbaiah et al. [19], Srinivasa Rao et al. [14] have developed and analyzed

economic production quantity models with weibull rate of decay and constant rate of production. Sridevi et al. [12],

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Srinivasa Rao et al. [16] have developed and analyzed economic production quantity models with random

production having constant decay.

In all these models, the production rate considered to be independent of stock on hand. But in reality if the product

is produced without considering the on hand stock the situation may lead excess inventories or heavy shortages.

Therefore it is the common practice in several of the industries dealing with perishable items cements, chemicals

etc., the production rate is adjusted depending on the stock on hand i.e. if more stock is there then the production rate

is reduced and if the less stock is there the production rate is increased. This type of production rate is known as

stock dependent production. Very little work has been reported in literature regarding economic production quantity

models with stock dependent production. Hence in this paper an economic production quantity model with stock

dependent production rate and time dependent demand having Pareto decay is developed and analyzed. The Pareto

decay is capable of characterizing the life time of the commodities having asymmetrically distributed variates. Its

instantaneous rate of decay is inversely propositional to ageing. Assuming shortages are allowed and fully

backlogged the model is developed. It is also developed to the case of without shortages.

The rest of the paper is organized as follows: section 2 deals with notations and assumptions and section 3

development of the EPQ model. In section 4, the optimal operating policies of the model are derived. In section 5, a

case study dealing with cement industry is presented. Section 6, is concerned with sensitivity analysis of the model

with respect to parameters and costs. Section 7, deals with the EPQ model without shortages. Section 8 is to discuss

the conclusions and scope for further work.

II. NOTATIONS AND ASSUMPTIONS OF THE MODEL

2.1 Notations

A Set up cost

𝜋 Shortage cost per unit per time

D (t) Demand rate

EPQ Economic Production Quantity

h Inventory holding cost per unit time

H Total inventory holding cost in a cycle time

I (t) Inventory level at any time T

c Unit production cost of the item

Q Production Quantity

Maximum inventory level

Maximum Shortage level

R (t) Rate of production at any time t

s Selling price of the items

Total Shortage cost in a cycle time

Time Point at which Production stops

(Production down time)

Time Point at which shortages begins

Time Point at which Production resumes (Production up time)

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T Production cycle time

TC Total production cost per unit time

TP Profit rate per unit time

TR Total revenue of the system pr unit time

b Deterioration rate parameter

(r, n, 𝜏, a, d) Demand rate parameters

(𝜂, k) Production rate parameters

2.2 Assumptions

i. Life time of the commodity is random and follows a Pareto distribution having probability density function

of the form

The instantaneous rate of deterioration is

ii. The demand is known and the demand rate is time dependent i.e. of the form . This is known

as power demand pattern, where r is the fixed quantity and n is the parameter of power demand pattern, the

value of n may be any positive number.

iii. The rate of production is dependent on stock on hand and is of the form

,

where 𝜂 is a constant such that 𝜂 > 0, k is the stock dependent production rate parameter, 0 ≤ k ≤ 1. It is

assumed that R(t) ≥ D(t) at any time where replenishment takes place. If k=0, then it includes the finite rate

of production.

iv. There is no repair or replacement of deteriorated items.

v. The planning horizon is finite. Each cycle will have length T.

vi. Lead time is zero.

vii. The inventory holding cost per unit time (h), the shortage cost per unit per unit time (π), the unit production

cost per unit time (c) and set up cost(A) per cycle are fixed and known.

III. EPQ MODEL WITH SHORTAGES

3.1 Model formulation

Consider an inventory system for deteriorating items in which the life time of the commodity is random and follows a

Pareto distribution. Here, it is assumed that shortages are allowed and fully backlogged. In this model the stock level

for the item is initially zero. Production starts at time t=0 and continues adding items to stock until the on hand

inventory reaches its maximum level S1. At time t = 0 deterioration of the item starts and stock is depleted by

consumption and deterioration while production is continuously adding to it. At time t = t1, the production is stopped

and stock will be depleted by deterioration and demand until it reaches zero at time t = t2. As demand is assumed to

occur continuously, at this point shortage begin to accumulate until it reaches its maximum level of S2 at t = t3. At this

point production will resume meeting the current demand and clearing the backlog. Finally shortages will be cleared

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at time t = T. Then the cycle will be repeated indefinitely. These types of production systems are common in cement

industries where production rate is stock dependent.

The schematic diagram representing the inventory system is shown in fig. 1

Inventory level I (t)

Fig.1 Schematic diagram representing the inventory level of the system with- shortages.

The differential equations governing the system in the cycle time (0, T) are;

With the initial conditions, I (0) = 0, I (t2) = 0 and I (T) = 0,

Solving the differential equations (1) to (4) the on hand inventory levels at time t are respectively.

The total inventory in the time period 0 ≤ t ≤ t1 is

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where, g (t, b, k) and f(t, b, k) are defined as in equation (6)

The total inventory in the period (t1, t2) is

Since I (t) is continuous at t1, equating (5) and (7) to establish the relationship between t1 and t2.

The maximum inventory level S1= I (t1) is

where, g(t1, b, k) and f(t1, b, k) are defined as in equation (13)

Similarly, since I (t) is continuous at t3 equating equations (8) and (9) to establish the relationship between t2 and t3,

therefore

The maximum shortage level S2 =I (t3) is

The backlogged demand is

Stock loss due to deterioration at time t is

where, g (t, b, k) and f (t, b, k) are as defined in equation (6)

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Total production in the cycle time (0, T) is

where, g (t, b, k) and f(t, b, k) are defined as in equation (6)

Let TC (t1, t3, T) is the total cost per unit time. Then, TC (t1,t3, T) is the sum of the setup cost per unit time, the

production cost per unit time, inventory holding per unit time and the shortage cost per unit time i.e.,

Holding cost in a cycle time is

Shortage cost in a cycle time is

Therefore, the total cost per unit time is

This implies,

where, g (t, b, k) and f(t, b, k) are as defined in equation (6)

3.2 Optimal Operating Policies of the Model

In this section, the optimal policies of the inventory system developed in section (3.1) are derived. To find the optimal

values of production down time t1 and production up time t3 one has to minimize the total cost TC (t1, t3,T) in

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equation (24) with respect to t1 and t3 and equate the resulting equations to zero. The condition for the solutions to be

optimal (minimum) is that the determinant of the Hessian matrix is positive definite i.e.

Differentiating TC (t1, t3, T) with respect to t1 and equating it to zero implies

where, g (t1, b, k) and f (t1, b, k) are as defined in equation (13)

Differentiating TC (t1, t3, T) with respect to t3 and equating it to zero implies

Solving the equations (26) and (27) simultaneously using numerical methods one can obtain the optimal values of t1

and t3. Substituting these optimal values of t1 and t3 in the equation (15), (19) and (24) one can get the optimal values

of , production quantity Q and total cost TC (t1, t3.T) respectively. For each set of optimal values, the determinant of

the Hessian matrix is computed and verified for positive semi definiteness.

3.3 Numerical Illustration

To expound the model developed, consider the case of deriving and economic production quantity, production down

time and production up time for a cement industry. Here the product is of a deteriorating type and has a random life

time which is assumed to follow Pareto distribution. Form the records and discussions held with the production and

marketing personnel the values of various parameters are considered. For different values of the parameters and costs,

the optimal values of production down time, production up time, optimal production quantity and total cost are

computed and presented in Table 1.

From Table 1, it is observed that when increase in deterioration parameter b from 1.1 to 1.5 units results a decrease in

production down time , production quantity Q* and an increase total cost TC* i.e. from 2.393 to 2.015 months,

Q* from 189.24 to 187.172 units and total cost TC* from

335.323 to 352.088. There is a slight decrease in production up time , from 10.951 to 10.722 months. When the

demand rate τ increases then the optimal production down time , production up time , production quantity Q* and

total cost TC* are increasing. The increase in holding cost h results a decrease in production down time from 2.319

to 2.017 months, production up time, from 11.001 to 10.701 months and increase in production quantity Q* from

183.417 to 187.781 units and increase in total cost TC* from ` 319.54 to `358.885. The increase in shortage cost π

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has significant effect on all optimal policies viz. production down time from 1.988 to 2.940 months, production up

time from 10.76 to 11.322 months, production quantity Q* from 182.897 to 202.281 units and total cost TC* from

` 304.398 to `357.851. The increase in production rate parameter k results an increase in production down time,

decrease in production up time, production quantity and increase in total cost i.e. Production quantity Q* from

222.058 to 186.715 and total cost from ` 341.124 to `342.588.

3.4 Sensitivity analysis

To study the effect of changes in the parameters and costs on the optimal values of production down time,

production up time and production quantity sensitivity analysis is performed taking the values A = `100, c = ` 4, h =

` 5, T = 12 months, π = ` 5, r = 200 units, b = 1.2, k = 0.4, η = 60.

Sensitivity analysis is performed by changing the parameters by -15%, -10%, -5%, 0%, 5%, 10% and 15%. First

changing the value of one parameter at a time while keeping all the rest at fixed values and then changing the values

of all the parameters simultaneously, the optimal values of t1, t3, Q and TC are computed and the results are presented

in Table 2. The relationship between parameters, costs and the optimal values are shown in fig 2.

From Table 2, it is observed that the demand rate τ and deteriorating parameter b have significant effect on

production down time, production up time, production quantity and total cost. Decrease in unit cost c results increase

in Q* from 183.221 to 193.049 units and decrease in TC* from ` 364.810 to ` 314.671. The increase in production

rate parameter η increases the production quantity Q*and total cost TC* from 162.866 units to 212.126 units and `

323.045 to ` 387.294. The increase in shortage cost results in decrease in production quantity Q* from 224.753 units

to 170.239 units and increase in total cost from ` 335.244 to ` 359.132.

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Fig. 2 Relationship between optimal values and parameters

IV. EPQ MODEL WITHOUT SHORTAGES

4.1 Model formulation

Consider a production system in which the production starts at time t = 0 and inventory level gradually increases

with the passage of time due to production and demand during the time interval (0, t1). At time t1 the production

is stopped and let S1 be the inventory level at that time. During the time interval (t1, T) the inventory decreases partly

due to demand and partly due to deterioration of items. The cycle continues when inventory reaches zero at time t =

T. The schematic diagram representing the model is shown in fig 3.

Fig 3 The schematic diagram representing the inventory level of the system without shortages

The differential equations governing the system in the cycle time (0, T) are;

with the boundary conditions I (0) = 0 and I (T) = 0, Solving the differential equations (28) and (29)

The instantaneous state of inventory at any time t during the interval (0, t1) is obtained as

where, g (t, b, k) and f (t, b, k) are as defined in equation (6)

The instantaneous state of inventory at any time t during the interval (t1, T) is obtained as

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The total inventory in the time period 0 ≤ t ≤ t1 is

where, g (t, b, k) and f (t, b, k) are as defined as in equation (6)

and

the total inventory in the time period t1 ≤ t ≤ T is

The maximum inventory level S1=I (t1) is

where, g (t1, b, k) and f (t1, b, k) are as defined in equation (13)

Stock loss due to deterioration at time t is

(36)

where, g (t, b, k) and f (t, b, k) are as defined in equation (6)

Total production quantity in the cycle time (0, T) is

where, g (t, b, k) and f (t, b, k) are as defined in equation (6)

Let TC (t1, T) is the total cost per unit time. Then, TC (t1, T) is the sum of the setup cost per unit time, the production

cost per unit time and inventory holding cost per unit time i.e.,

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Holding cost in the cycle time T is

Therefore the total cost per unit time is

This implies,

(41)

where, g (t, b, k) and f (t, b, k) are as defined as in equation (6)

4.2 Optimal Operating Policies of the Model

In this section, we obtain the optimal policies of the inventory system developed in section (4.1). The problem is to

find the optimal values of production down time t1 that minimize the total cost TC (t1, T) over (0, T). To obtain the

optimal values we differentiate TC (t1, T) in equation (41) with respect to t1 and equate the resulting equation to zero.

The condition for the solutions to be optimal (minimum) is that

Differentiating TC (t1, T) with respect to t1 and equating to zero one can get

where, g (t1, b, k) and f (t1, b, k) are as defined as in equation (13)

Solving the equation (44) using numerical methods one can get the optimal value of t1. Substituting the optimal value

of t1 in the equation (37) and (41) the optimal values of production quantity Q and total cost TC can be obtained

respectively.

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4.3 Numerical Illustration

To expound the model developed, consider the case of deriving an economic production quantity and production

down time for a cement manufacturing unit. Here, the product is deteriorating type and has random life time and

assumed to follow a Pareto distribution. Based on the discussions held with the personnel connected with the

production and marketing and the records it is observed that the deterioration parameter b is estimated to vary from

1.2 to 1.6 months and demand parameter r to vary from 200 to 300 units respectively. The other parameters are

considered as T = 12 months, c = ` 3 to ` 5, h = ` 4 to ` 6, k = 0.4 to 0.8 and 𝜂 = 50 to 70. Substituting the values of

the parameters and costs in the equation (44) and solving numerically, the optimal values of production down

time , production quantity Q* and total cost TC* are obtained and are presented in Table 3.From Table 3, It is

observed that the increase in deterioration parameter b from 1.2 to 1.6 has shown an increasing trend in production

down time from 7.264 to7.616 months, production quantity Q* from 299.569 to 320.422 units and total cost TC*

from `267.290 to `267.436. Whereas an increase in demand parameter r from 200 to 300 results as a increase in

optimal values of from 6.574 to 7.821 months, Q* from 265.933 to 331.535 units and TC* from ` 257.734 to Rs

272.31.

The increase in unit cost c from ` 3 to ` 5 has a decreasing effect on

and Q* an increasing effect on TC* viz.

Production down time from 7.454 to 7.078 months, production quantity Q* from 304.117 to 294.610 units, total

cost TC*, from ` 242.117 to ` 292.047. The increase in holding cost h from `4 to ` 6 results an increase in optimal

values , Q* and TC* i.e. production down time from 7.078 to 7.39 months production quantity Q* from

294.610 to 302.883 units and total cost TC* from `235.304 to `298.998

The increase in production rate parameter k from 0.4 to 0.8 results an increase in optimal values

and decrease in

production quantity Q* and total cost TC* i.e. production down time from 6.669 to 7.763 months, production

quantity Q* from 307.428 to 292.277 units and total cost TC* from ` 292.999 to 245.862. Whereas the increase in

production rate parameter 𝜂 from 50 to 70 results a decrease in production down time from 7.821 to 6.787

months, increase in production quantity Q*276.279 to 321.717 units and total cost TC* from ` 228.314 to ` 303.164

4.4 Sensitivity Analysis

To study the effects of changes in the parameters on the optimal values of production down time and production

quantity a sensitivity analysis is performed taking the values of the parameters as b = 1.2, n = 2, r = 250, c = `4/-,

h = `5/-, k=0.6, A = 100 and η = 60.Sensitivity analysis is performed by changing the parameter values by -15%, -

10%, -5%, 5%, 10% and 15%. First changing the value of one parameter at a time while keeping all the rest at fixed

values and then changing the values of all the parameters simultaneously, the optional values of production down

time, production quantity and total cost are computed. The results are presented in Table 4. The relationship between

parameters, costs and the optimal values are shown in Fig 4.

From Table 4, It is observed that variation in the deterioration parameters b has considerable effect on production

down time, optimal production quantity and total cost i.e. production down time from 7.079 to 7.432 months,

production quantity Q* from 289.375 to 309.27 units and total cost TC* from ` 267.037 to Rs 267.411 Similarly

variation in demand parameter r has slight effect on production down time from 6.761 to 7.692 months and

moderate effect on production quantity Q* from 274.537 to 323.658 units and total cost TC* from ` 260.66 to `

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271.389. The decrease in unit cost c results an increase in production down time from 7.152 to 7.378 months,

optimal production quantity Q*, from 296.593 to 302.569 units and decrease in total cost TC* ` 282.194 to

`252.236 The increase in production rate parameters k and η results slight variation in production quantity Q*from

303.068 to 296.19 units and from 278.659 to 319.563 units and total cost TC* from ` 278.322 to `257.134 and from

` 232.366 to ` 299.702 respectively. The increase in holding cost h has significant effect on optimal values

production down time from 7.132 to 7.363 months, production quantity Q* from 296.058 to 302.176 units and

total cost TC* from ` 243.337 to ` 291.089. When all the parameters change at a time that highly effects on optimal

values i.e. production down time from 7.252to 7.296 months, production quantity Q* from 259.597 to 338.122

units and total cost TC* from ` 209.856 to ` 330.065 respectively.A comparative study of with and without

shortages revealed that allowing shortages has significant influence on optimal production schedule and total cost.

This model includes some of the earlier inventory models for deteriorating items with Pareto decay as particular

cases for specific values of the parameters When k = 0, this model includes EPQ model for deteriorating items with

Pareto decay and constant rate of demand and finite rate of replenishment. When b = 0, this model becomes EPQ

model with stock dependent production and time dependent demand. When n=1, this model includes EPQ model

with stock dependent production and constant rate of demand.

V/. CONCLUSIONS

This paper introduces an EPQ model with a stock dependent production rate and time dependent demand having

Pareto decay. The stock dependent production will avoid wastage in excess inventories and loss due to shortages.

The Pareto rate of decay can well characterize the life time of the commodities like cement which exhibit high rate

of decay in the early periods and slow rate of decay after certain period. That is the lifetimes are left skewed having

long upper time distribution. The optimal production down time, production uptime and production quantity are

derived using MathCAD code and Hessian matrix. A case study dealing with perishable item like cement has

demonstrated that the stock dependent production rate has significant influence on total production cost.

The sensitivity analysis of the model reveals that the deterioration parameters and demand parameters have

tremendous influence on optimal values of the production down time and up time. With historical data on lifetime

and demand the production manger can estimate the parameters of the model can produce optimal production

quantity. This model can also be extended with inflation and delay in payments which will be taken elsewhere. For

specific values of the parameters this model provides spectra of models which are useful for scheduling a variety of

production processes.

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Table 1 OPTIMAL VALUES OF t1, t3, Q and TC for different values of the parameters and

costs for the model-without shortages

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Table 2 Sensitivity Analysis of the model with –shortages

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Table 3 OPTIMAL VALUES OF t1, Q and TC for different values of the parameters and costs

for the model-without shortages

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Table 4 Sensitivity Analysis of the model-without shortages

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Fig 4; Relationship between optimal values and parameters

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